LEARN · EXPLAIN · REVISE

Find the derivative of e^(−x).

Read the idea, work independently, then explain what changed.

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高二選擇性必修第二册(A版).pdf · 5.2 · PDF 77 / printed page 72

Revisit first: Rates and tangent meaning

TOPIC 01

Derivative rules and domains

Choose power, product, quotient or chain rules from the expression structure, retaining domains.

What you will be able to explain

Choose power, product, quotient or chain rules from the expression structure, retaining domains.
Recover an unknown inner rate and audit missing derivative factors.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Standard#Your turn

Find the derivative of e^(−x).

f=e^{-x}

Use the stated real domain and distinguish stationary candidates, attained extrema and actual concavity changes.

Skills and prerequisite lessons

Scope: School extension · This task develops the school textbook skill beyond the explicitly listed JAE syllabus topics.

Working and explanation

BUILD THE REASONING

Hint 1

The inside function −x has derivative −1.

Hint 2

Start from this intermediate relation.

f'=e^{-x}(-1)

Worked solution

The inside function −x has derivative −1.
Establish and simplify this mathematical relation.
$f'=e^{-x}(-1)$
Establish and simplify this mathematical relation.
$f'=-e^{-x}$
The exponential remains positive although its rate here is negative.

f′=−e^(−x).

Checks and common pitfalls: The exponential remains positive although its rate here is negative.

Reasoning checklist · self / teacher assessment

The inside function −x has derivative −1.
Give the intermediate relations and explain the conclusion with its restrictions.
The exponential remains positive although its rate here is negative.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Predict	Predict the derivative of x·x before expanding.
8	Build the idea	Sample response: each factor contributes 1·x, so the sum is 2x.
10	Guided comparison	Sample response: F′(2)=4g′(4)=24, so g′(4)=6.
12	Independent practice and discussion	Sort tasks by their outermost operation, then contrast polynomial and extension domains.
5	Exit and next step	Exit: (3x+1)² differentiates to 6(3x+1), and ln(3x−2) needs x>2/3.

Question sequence

Predict the derivative of x·x before expanding.
Sample response: F′(2)=4g′(4)=24, so g′(4)=6.

Board development

Differentiate the structure
$(uv)'=u'v+uv';\ (f\circ g)'=f'(g)g'$

Expected correct and incorrect responses

Correct sample: 6
Anticipated wrong response: 2 — Omits the inner derivative in a composition.
Anticipated wrong response: 6x³ — Multiplies derivatives instead of adding the two product-rule contributions.
Omitting the inner derivative.
Multiplying or dividing derivatives as if the original arithmetic rules still apply.

Follow-up questions

Follow-up: Keep both the outer derivative and inner derivative. Then explain why 2 fails.
Follow-up: First combine the powers, then differentiate. Then explain why 6x³ fails.
At k=0, explain why the function is constant and its derivative vanishes.

Remediation

Sample response: each factor contributes 1·x, so the sum is 2x.
Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.

Exit check

Exit: (3x+1)² differentiates to 6(3x+1), and ln(3x−2) needs x>2/3.

Assessment checklist and sample responses

Representation and conditions
Both factors are differentiable; (uv)′=u′v+uv′.
Connected mathematical reasoning
For (x²+1)(x−2), derivative 2x(x−2)+(x²+1)=3x²−4x+1.
Interpretation and boundary check
ln(3x−2) needs x>2/3, not merely x≠2/3.